221 problems

Let $G$ be a finite graph, let $\lambda(G)$ denote the spectral radius of its adjacency matrix, and let $G_p$ be obtained from $G$ by retaining each edge independently with probabi…

Higher-dimensional max-flow min-cut conjecture. There exists a map $\mu$ on the unit sphere such that, for every convex set $A$ with $0$ in its interior,

The bunkbed conjecture for transitive tournaments. The statement

Consider the infinite connected components of the corner-percolation configuration, each assigned its integer-valued level. Level-bijection conjecture. The function which maps each…

Let $(G_n)_n=(V_n,E_n)_n$ be a sequence of finite, connected, vertex-transitive graphs with vertex degree $\bDelta_n$ diverging as $n\to\infty$. For a fixed sequence and fixed $u$,…

Monotonicity conjecture. For $\lambda<\lambda'\in(0,1]$,

Häggström–Pemantle's conjecture. In every dimension $d\geq 2$, infinite coexistence has positive probability if and only if the infection intensities are equal:

Let $ell geqslant 2$ and let ${\mathcal R}$ be an $ell$-vertex size rule. Define $t_{\mathrm{b}}^{{\mathcal R}}$ as the supremum of the set of $t \geqslant 0$ for which the suscept…

Let $G$ and $G'$ be groups with Cayley graphs whose asymptotic cones are isometric. A group is non-elementary hyperbolic if it is hyperbolic and is not virtually cyclic. Sapir's co…

Commuting infinite subgroups conjecture. The group $G$ contains two infinite subgroups $H_1$ and $H_2$ which commute with each other:

Consider the competition process on $\mathbb{Z}^d$ with blue and yellow infections, whose transmission parameters are $p_{\text{b}}$ and $p_{\text{y}}$, respectively. Let $p_c$ be…

Corrector Gaussian free field conjecture. Let $d\ge1$. Then the law of

High-dimensional corrector tightness conjecture. Let $d\gg1$. Then for each $\epsilon>0$ there exists $K<\infty$ such that

Theorem 5.1 conjecture. Theorem 5.1 is true in all $d\ge2$.

Positive critical regrowth conjecture. The critical regrowth parameter is positive:

Supercritical upper-bound conjecture. The displayed limsup is finite.

Unoriented percolation scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that the scaling relation (rhoscal) holds for $\rho^{(\mathrm{pe})}$.

Let $G$) be a graph with distinguished vertex $x$, and let $H=G\times\{0,1\}$ have two copies of $G$, with corresponding vertices joined by an edge. Run Richardson's model on $H$ f…

Let $a=\log q$ for $q\in(0,1)$, and let $\mathbb{P}(N)(q)$ denote the probability of the annular crossing event $N$ described above. Set … Here $\eta$ denotes Dedekind's eta functi…

Let $p\in(p_c,1)$, where $p_c=\frac{1}{2}$ is the critical probability for bond percolation on $\mathbb{Z}^2$. For the biased random walk on the infinite percolation cluster, let…

Let $B_r$ be the ball of radius $r$ around the root in the type II uniform infinite planar triangulation, and let $\overline{B}_r$ be its hull, obtained by adjoining to $B_r$ all f…

Let $G_n$ be a sequence of finite 3-regular graphs with growing girth, converging locally to the 3-regular tree $T_3$, and assume that the thresholds $p^{G_n}$ remain bounded away…

Let $G$ be a finite graph and define … Here $\partial S$ is the vertex boundary used in the paper, and $p^G$ is the percolation threshold. Isoperimetric-threshold conjecture. If…

Let $G$ be a finite transitive graph. Say that $G$ has the $EIT(c)$ property, for $0<c<1$, if for every pair of vertices $u,v\in G$ there is a set of paths from $v$ to $u$ equipped…

Let $G$ be a finite transitive graph, let $|G|$ denote its number of vertices, and let $p^G$ be its percolation threshold. Diameter criterion conjecture. There is a constant $C<1$…